Solve the simultaneous equations using both elimination and substitution method.

Answer:

option A

f(x) = (x – 1)2 + 3

Step-by-step explanation:

Given in the question a function,

f(x) = 4 + x² – 2x

f(x) = 4 + x² – 2x

here a = 1

        b = -2

        c = 4

x = -b/2a

h = -(-2)/2(1)

h = 2/2

h = 1

Step 3

Find k

k = 4 + 1² – 2(1)

k = 3

Step 4

To convert a quadratic from y = ax² + bx + c form to vertex form,

y = a(x - h)²+ k

y = 1(x - 1)² + 3

y = (x - 1)² + 3

ANSWER

The vertex form is

[tex]f(x) = {(x - 1)}^{2} + 3[/tex]

EXPLANATION

The given function is

[tex]f(x) = 4 + {x}^{2} - 2x[/tex]

This is the same as:

[tex]f(x) = {x}^{2} - 2x + 4[/tex]

We add and subtract the square of half the coefficient of x.

[tex]f(x) = {x}^{2} - 2x + {( -1 )}^{2} - {( -1 )}^{2} + 4[/tex]

[tex]f(x) = {x}^{2} - 2x + 1 - 1 + 4[/tex]

The first three term is a perfect square trinomial:

[tex]f(x) = {(x - 1)}^{2} + 3[/tex]

The vertex form is

[tex]f(x) = {(x - 1)}^{2} + 3[/tex]

Answer:

r = 6

Step-by-step explanation:

Given that M is directly proportional to r³ then the equation relating them is

M = kr³ ← k is the constant of proportionality

To find k use the condition r = 4 when M = 160

k = [tex]\frac{M}{r^3}[/tex] = [tex]\frac{160}{64}[/tex] = 2.5, so

M = 2.5r³ ← equation of proportionality

When M = 540, then

540 = 2.5 r³ ( divide both sides by 2.5 )

216 = r³ ( take the cube root of both sides )

r = [tex]\sqrt[3]{216}[/tex] = 6

Answer:

B

Step-by-step explanation:

Find all probabilities:

A. False

[tex]Pr(\text{red shirt}|\text{large shirt})=\dfrac{\text{number red large shirts}}{\text{number large shirts}}=\dfrac{42}{77}=\dfrac{6}{11}\\ \\Pr(\text{large shirt})=\dfrac{\text{number large shirts}}{\text{number shirts}}=\dfrac{77}{165}=\dfrac{7}{15}[/tex]

B. True

[tex]Pr(\text{blue shirt}|\text{large shirt})=\dfrac{\text{number blue large shirts}}{\text{number large shirts}}=\dfrac{35}{77}=\dfrac{5}{11}\\ \\Pr(\text{blue shirt})=\dfrac{\text{number blue shirts}}{\text{number shirts}}=\dfrac{75}{165}=\dfrac{5}{11}[/tex]

C. False

[tex]Pr(\text{shirt is medium and blue})=\dfrac{\text{number medium and blue shirts}}{\text{number shirts}}=\dfrac{48}{165}=\dfrac{16}{55}\\ \\Pr(\text{medium shirt})=\dfrac{\text{number medium shirts}}{\text{number shirts}}=\dfrac{88}{165}=\dfrac{8}{15}[/tex]

D. False

[tex]Pr(\text{large shirt}|\text{red shirt})=\dfrac{\text{number red large shirts}}{\text{number red shirts}}=\dfrac{42}{90}=\dfrac{7}{15}\\ \\Pr(\text{red shirt})=\dfrac{\text{number red shirts}}{\text{number shirts}}=\dfrac{90}{165}=\dfrac{6}{11}[/tex]

Answer:

[tex]7.7n=112.42[/tex]

[tex]n=14.6[/tex]

Step-by-step explanation:

The product is the result obtained by multiplying two factors. Then, the sentence ""The product of a number n and 7.7 equals 112.42" can be expressed with the following equation:

[tex]7.7n=112.42[/tex]

To solve for the variable "n", you has two apply the Division property of equality and divide both sides of the equation by 7.7

Therefore, the value of "n" is:

[tex]\frac{7.7n}{7.7}=\frac{112.42}{7.7}\\\\n=14.6[/tex]

Answer:

1)no

2)no

Fractions are rational numbers

Step-by-step explanation:

irrational numbers are numbers which can't be written as ratios, fractions, or rational decimals

hope this helps!

Answer:

a

Step-by-step explanation:

1/6p + (-4/5) is the equivalent expression. You have to add like terms, meaning constants are added to constants, variables are added to variables, etc. the result you get from adding like variables leaves you with 1/6p + (-4/5) or 1/6p - 4/5

Answer:

[tex]\alpha=44\°[/tex]

Step-by-step explanation:

By definition, the tangent of an angle is the quotient between the side opposite the angle and the side adjacent to the angle

In other words:

[tex]tan(\alpha) = \frac{opposite}{adjacent}[/tex]

In this triangle, the length of the side adjacent to the desired angle is 50, and the length of the opposite side is 48

So:

[tex]tan(\alpha) = \frac{48}{50}\\\\tan(\alpha)= 0.96[/tex]

Finally

[tex]\alpha =arctan(0.96)\\\\\alpha=44\°[/tex]

Answer:

Final answer is [tex]?=44[/tex] degree.

Step-by-step explanation:

Using given information in the picture, we need to find the missing value of angle "?"

Apply formula of tangent function which is :

[tex]\tan\left(\theta\right)=\frac{opposite}{adjacent}[/tex]

[tex]\tan\left(?^o\right)=\frac{48}{50}[/tex]

[tex]\tan\left(?^o\right)=0.96[/tex]

[tex]?=\tan^{-1}\left(0.96\right)[/tex] degree

[tex]?=43.830860672092581097187030418859[/tex] degree

Hence final answer is [tex]?=44[/tex] degree.

By buying 9 packs of 4 cartons and 2 single cartons, Lewis was able to spend the least amount of money to get the 38 cartons he needed. The total cost was £14.94.

The subject of this question is mathematics, specifically cost optimization and arithmetic. Lewis can buy single carton of juice for 45p or a pack of 4 cartons for £1.56. The goal is to minimize the cost. First, Lewis can start by buying packs of 4. The number of 4-carton packs he can buy is 38 divided by 4 which gives you 9 packs that include 36 cartons. He then needs to buy 2 more single cartons to reach 38. So, the total cost is 9 packs times £1.56/pack plus 2 single cartons times 45p/carton. Doing the arithmetic:

Convert 90p to pounds that is £0.90. Adding both gives £14.94. Therefore, Lewis spent a total of £14.94 on juice cartons.

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Answer:

x= -5 makes the given equation  -4x = 20 true

Step-by-step explanation:

We have been given the equation;

-4x = 20

we are to determine the value of x that will make the equation true. We simply make x the subject of the formula;

divide both sides of the equation by -4;

(-4x)/(-4) = 20/(-4)

x = -5

Therefore, x= -5 makes the given equation  -4x = 20 true

Answer:

[tex]x=-5[/tex]

Step-by-step explanation:

The given linear equation is:

-4x = 20

To find the value of x that makes the equation true, we need to solve for x.

We divide both sides by -4 to obtain:

[tex]\frac{-4x}{-4}=\frac{20}{-4}[/tex]

This is the sane as:

[tex]\frac{-4x}{-4}=\frac{-5\times-4}{-4}[/tex]

We cancel out the common factors to get:

[tex]x=-5[/tex]

For this case we must add the given series. We expanded the series for each of the values of i, that is, from 1 to 5:[tex]3 (1) +3 (2) +3 (3) +3 (4) +3 (5) =\\3 + 6 + 9 + 12 + 15 =[/tex]

We add the terms:

[tex]9 + 9 + 12 + 15 =\\18 + 12 + 15 =\\30 + 15 =\\45[/tex]

Finally, the value of the given series is 45.

Answer:

45

Answer:

The domain = (-∞ , -1/4) ∪ (-1/4 , ∞)

The range = (-∞ , 21/4)∪(21/4 , ∞)

The answer is not in the choices

Step-by-step explanation:

* Lets revise how to find the inverse function

- At first write the function as y = f(x)

- Then switch x and y

- Then solve for y

- The domain of f(x) will be the range of f^-1(x)

- The range of f(x) will be the domain of f^-1(x)

* Now lets solve the problem

- At first find the domain and the range of f(x)

∵ f(x) = (x - 9)/(21 - 4x)

- The domain is all real numbers except the value which

  makes the denominator = 0

- To find this value put the denominator = 0

∴ 21 - 4x = 0 ⇒ subtract 21 from both sides

∴ -4x = -21 ⇒ ÷ -4 both sides

∴ x = 21/4

∴ The domain = R - {21/4} OR the domain = (-∞ , 21/4)∪(21/4 , ∞)

* Now lets find the range

- The range will be all the values of real numbers except -1/4

 because the horizontal asymptote equation is y = -1/4

- To find the horizontal asymptote we find the equation y = a/b

 where a is the coefficient of x up and b is the coefficient of x down

∵ The coefficient of x up is 1 and down is -4

∴ The equation y = 1/-4

∴ The value of y = -1/4 does not exist

∴ The range = R - {-1/4} OR the range = (-∞ , -1/4) ∪ (-1/4 , ∞)

* Switch the domain and the range for the f^-1(x)

∴ The domain = (-∞ , -1/4) ∪ (-1/4 , ∞)

∴ The range = (-∞ , 21/4)∪(21/4 , ∞)

Answer:

Step-by-step explanation:

Just use the slope equation.

Answer:

B. 1.47*10^18

Step-by-step explanation:

Given

Distance of largest start from earth in light years = 250000

1 light year = 5880000000000 miles

So,

Distance of largest star from earth in miles = 250000 * 5880000000000

= 1470000000000000000 miles

In order to convert in scientific notation, point will be moved 18 places to the left, so the number will become:

1.47* 10^18 miles

So option B is correct..

Answer: OPTION B

Step-by-step explanation:

Scientific notation has the form:

[tex]a10^n[/tex]

Where "a" is a number between 1 and 10 but is not less than 10 and "n" is an integer.

If a light-year is about 5,880,000,000,000 miles and the large star is 250,000 light-years from Earth, then this distance in miles is:

[tex]d=(250,000)(5,880,000,000,000)\\d=1,470,000,000,000,000,000.0\ mi[/tex]

To express this distance in scientific notation, the decimal point must be after the first digit. You can observe that the decimal point must be moved 18 places, then:

[tex]a=1.47[/tex] and  [tex]n=18[/tex]

Therefore, you get that the distance of the large star from Earth in scientific notation is:

[tex]d=1.47*10^{18}\ mi[/tex]

The complete factorization of the polynomial [tex]x^3 + x^2 + 9x + 9[/tex] is [tex]x + 1)(x^2 + 9)[/tex] using real coefficients, and [tex](x + 1)(x + 3i)(x - 3i)[/tex] using complex coefficients.

The complete factorization of the polynomial [tex]x^3 + x^2 + 9x + 9[/tex] can be found by grouping and factoring terms with common factors.

To proceed, let's group the terms as follows: [tex](x^3 + x^2) + (9x + 9).[/tex]

Factoring out the common factors in each group gives us [tex]x^2(x + 1) + 9(x + 1).[/tex]

Now, we notice that (x + 1) is a common factor in both terms, so we can factor it out to get our complete factorization: [tex](x + 1)(x^2 + 9).[/tex]

As [tex]x^2 + 9[/tex] can't be factored further using real coefficients, this is the fullest factorization in the realm of real numbers.

However, if complex numbers are considered, [tex]x^2 + 9[/tex] can be expressed as [tex](x + 3i)(x - 3i)[/tex], where i is the imaginary unit.

So, the complete factorization using complex coefficients is [tex](x + 3i)(x - 3i)[/tex]

Answer:

The probability is 1/15 or 6.67%

Step-by-step explanation:

step 1

Find the probability that when she shakes the piggy bank a half-dollar fall out

P=2/15

step 2

Find the probability that when she shakes the piggy bank a dime fall out

P=7/14

step 3

Find he probability that when she shakes the piggy bank a half-dollar and then a dime fall out

P=(2/15)(7/14)=1/15

Convert to percentage

(1/15)*100=6.67%

Answer:

Simplifying

5x = 3x + 14

Reorder the terms:

5x = 14 + 3x

Solving

5x = 14 + 3x

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3x' to each side of the equation.

5x + -3x = 14 + 3x + -3x

Combine like terms: 5x + -3x = 2x

2x = 14 + 3x + -3x

Combine like terms: 3x + -3x = 0

2x = 14 + 0

2x = 14

Divide each side by '2'.

x = 7

Simplifying

x = 7

Step-by-step explanation:

This is the answer (I believe): 18 + x = 29

Answer:

11

Step-by-step explanation:

29-18 is 11

Answer:

212°

Step-by-step explanation:

We have the measure of CE and the measure of ED, so all we have to do is combine those two to get the measure of CED.

The arc CE measures 52 degrees.

The arc ED measures 160 degrees

The arc CED is the sum of arc CE and ED, so:

CED = 52° + 160° = 212°

We don't have any length measurement inside that circle to calculate the length of the arc CED in inches, centimeters or otherwise... so we can only measure it degrees.

No, to be proportional, there would be a consistent pattern between x and y. For example 2 times 2.5 is 5. And 3 times 2 is 6. The ratio is not the same.

Answer:

the first one is 23

the second one is 70-79

the third one is 11

Step-by-step explanation:

this is because frequency refers to the amount of people who got the scores on the x-axis.

so for the first one you would count up how many people participated i.e. 2+9+7+5 = 23

the second one is asking for the mode test score and mode refers to the most common, so the most common test score is the 70-79 bar

the third one is asking for how many people scored below 80, which is 11 as 2 people scored 60-69 and 9 scored 70-79 on the test (2+9=11)

Hope this helps :)

the answer is 23

hope this helps

let's recall that the volume of a rectangular prism is simply the product of its 3 dimensions.

so the volume for the first one is simply 3*4*8 = 96.

now, let's do some quick prime factoring of that 96

96 = 2*2*2*2*2*3

since we know the factors of 96, and we can rearrange them in any way we wish and will always have a product of 96, well, le'ts do some rearranging for the second prism.

the second prism has 4 and 4, namely 2*2 and 2*2, what's leftover? 2*3.

so the second prism must be (2*2) * (2*2) * (2*3), namely a 4x4x6.

For this case we have that the boundary line of inequality is dotted, therefore equality is not included.

We find the slope of the line, substituting two points that pass through it:

[tex](0,3)\\(3,5)\\m = \frac {5-3} {3-0} = \frac {2} {3}[/tex]

We evaluate a point within the region and see which case is met:

[tex](x, y) :( 0.6)[/tex]

We replace:

[tex]A) y <\frac {2} {3} x + 3\\6 <0 + 3\\6 <3[/tex]

It is not fulfilled, we discard the first option.

[tex]C) y> \frac {2} {3} x + 3\\6> 0 + 3\\6> 3[/tex]

Is fulfilled!

Answer:

Option C

Answer:

-10

Step-by-step explanation:

xy = k

(-2)(5) = k

-10 = k

Answer: -10

Step-by-step explanation:

Given: The inverse variation equation is given by :-

[tex]xy=k[/tex]

To find : The constant of variation, k, when x=-2 and y=5

Substitute the given values of x and y in the given inverse equation , we get

[tex]-2\times 5=k\\\\\Rightarrow\ k=-10[/tex]

Hence, the value of constant of variation = -10

The value of the expression is 14.

The answer is I. 8

Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.

y = 4x - 10. The linear functions y = 4x - 10 represent the line given by the point-slope equation y - 2 = 4(x - 3).

In order to solve this problem, we have to take the point-slope equation y - 2 = 4(x - 3) and convert it to the linear function form y = mx + b as follow:

y - 2 = 4(x - 3)

y - 2 = 4(x) - (4)(3)

y - 2 = 4x - 12

y - 2 + 2 = 4x -12 +2

y = 4x - 10 (Linear function)

Answer: The answer to your question is 200

Step-by-step explanation: because when you are multiply a percentage by a whole number, you are trying to find what percentage of the whole number makes the problem true. In your case, the problem would be set up like this:

44% of what number is 88

0.44 of x is 88

So, now that you have the problem set up, the final answer would be 200 after you have gone through each answer and substituted each value to come up with a true statement

Answer:

x = 8

Step-by-step explanation:

Given

- x = - 8 ( multiply both sides by - 1

- x × - 1 = - 8 × - 1, that is

x = 8

Answer:

x = 8

Step-by-step explanation:

Given

- x = - 8 ( multiply both sides by - 1

- x × - 1 = - 8 × - 1, that is

x = 8

Answer:

[tex]x_1=-4\\\\x_2=\frac{1}{3}[/tex]

Step-by-step explanation:

Given the quadratic equation:

[tex]2x^2+14x-4=-x^2+3x[/tex]

You can follow these steps in order to solve it:

- Make the equation equal to 0:

[tex]2x^2+14x-4+x^2-3x=0[/tex]

- Now, add the like terms:

[tex]3x^2+11x-4=0[/tex]

- Finally, apply the Quadratic formula ([tex]x=\frac{-b\±\sqrt{b^2-4ac} }{2a}[/tex]):

In this case we know that:

[tex]a=3\\b=11\\c=-4[/tex]

Therefore, substituting values into the Quadratic formula and evaluating, we get:

[tex]x=\frac{-11\±\sqrt{11^2-4(3)(-4)} }{2(3)}\\\\x_1=-4\\\\x_2=\frac{1}{3}[/tex]

To find the solutions to the quadratic equation 2x^2+14x-4=-x^2+3x, we need to rearrange the equation and use the quadratic formula.

The quadratic equation is 2x^2+14x-4=-x^2+3x. To find the solutions, we need to rearrange the equation in the form ax^2 + bx + c = 0. Combining like terms, we get 3x^2 + 11x - 4 = 0. Now we can use the quadratic formula to find the solutions:

x = (-b ± √(b^2 - 4ac)) / 2a

Plugging in the values a = 3, b = 11, and c = -4 into the formula, we can calculate the solutions.

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